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The developments over the last five decades concerning numerical discretisations of the incompressible Navier--Stokes equations have lead to reliable tools for their approximation: those include stable methods to properly address the…
We consider the modified Navier-Stokes equations in R3 describing the motion of a fluid in the presence of a rotating rigid body. Weighted Sobolev spaces are used to describe the behavior of solutions at large distances. Under suitable…
We investigate the long-time behavior of solutions to a stochastically forced one-dimensional Navier-Stokes system, describing the motion of a compressible viscous fluid, in the case of linear pressure law. We prove existence of an…
We investigate fractional stochastic Navier-Stokes equations in $d\ge 3$, driven by the random force $(-\Delta)^{\frac{\theta}{2}}\xi$ which, as we show, corresponds to a fractional version of the Landau-Lifshitz random force in the physics…
We study a modified three-dimensional incompressible anisotropic Navier-Stokes equations. The modification consists in the addition of a power term to the nonlinear convective one. This modification appears naturally in porous media when a…
In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier--Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically…
We investigate the Navier-Stokes turbulence driven by a stochastic random Gaussian force. Using a field-theoretic approach, we uncover an anomaly that brings hidden structure to the theory. The anomaly is generated by a non-self-adjoint…
Fully-developed incompressible Navier-Stokes turbulence in three dimensions is a dissipative dynamical system that exhibits strong departure from absolute equilibrium. Nevertheless, several kinds of representation by Tsallis equilibria have…
In recent literature several derivations of incompressible Navier-Stokes type equations that model the dynamics of an evolving fluidic surface have been presented. These derivations differ in the physical principles used in the modeling…
Since Kolmogorov's theory, turbulence has been studied using various methods, many of which could be now be understood in a probabilistic framework. Herein, a comprehensive review of the advances made on stochastic theory of turbulence…
Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years,…
The NS equation is considered (in 2 & 3 dimensions) with a fixed forcing on large scale; the stationary states form a family of probability distributions on the fluid velocity fields depending on a parameter R (Reynolds number). It is…
Turbulence is an ubiquitous phenomenon in natural and industrial flows. Since the celebrated work of Kolmogorov in 1941, understanding the statistical properties of fully developed turbulence has remained a major quest. In particular,…
In this paper, we give an overview of the results established in [3] which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system…
In order to address the difficulties of classical fluid kinematics in describing vorticity and the paradox of linear correlation between viscous force and vorticity in the Navier-Stokes equations, the study examines the inherent…
In the given paper, we confront three finite difference approximations to the Navier--Stokes equations for the two-dimensional viscous incomressible fluid flows. Two of these approximations were generated by the computer algebra assisted…
We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain…
In the present work, we investigate a numerical one-dimensional solver to the Navier-Stokes equation that retains all terms, including both pressure and dissipation. Solutions to simple examples that illustrate the actions of the nonlinear…
The continuity of the kinetic energy is an important property of incompressible viscous fluid flows. We show that for any prescribed finite energy divergence-free initial data there exist infinitely many global in time weak solutions with…
We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with $H^1$-conforming pressure…